Question

In the ground state of element 115, Uup, a. how many electrons have \(n=5\) as one of their quantum numbers? b. how many electrons have \(\ell=3\) as one of their quantum numbers? c. how many electrons have \(m_{\ell}=1\) as one of their quantum numbers? d. how many electrons have \(m_{s}=-\frac{1}{2}\) as one of their quantum numbers?

Short answer

a. There are 32 electrons with \(n=5\). b. There are 28 electrons with \(\ell=3\). c. There are 12 electrons with \(m_{\ell}=1\). d. There are 58 electrons with \(m_{s}=-\frac{1}{2}\).

Step-by-step solution

Determine which orbitals have n=5

For n=5, we have 5 possible orbitals, including s, p, d, f, and g, i.e., ℓ=0,1,2,3,4.

Calculate the number of electrons in n=5 orbitals

Let's count the electrons in each of these 5 orbitals: - 5s: 2 electrons (ℓ=0, 1 s orbital) - 5p: 6 electrons (ℓ=1, 3 p orbitals) - 5d: 10 electrons (ℓ=2, 5 d orbitals) - 5f: 14 electrons (ℓ=3, 7 f orbitals) - 5g: not occupied in element 115 (ℓ=4, 9 g orbitals) The total number of electrons in n=5 orbitals is 2+6+10+14=32 electrons. #b. Electrons with ℓ=3#

Identify which orbitals have ℓ=3

Orbitals with ℓ=3 are f orbitals. The first f orbital is 4f since there is no 1f, 2f, or 3f.

Calculate the number of electrons in f orbitals

Calculate the electrons in the 4f and 5f orbitals: - 4f: 14 electrons - 5f: 14 electrons The total number of electrons with ℓ=3 is 14+14=28 electrons. #c. Electrons with mℓ=1#

Identify which orbitals have mℓ=1

mℓ=1 is contained in two types of orbitals: p (mℓ=-1,0,1) and f (mℓ=-3,-2,-1,0,1,2,3).

Calculate the number of electrons with mℓ=1 in p and f orbitals

Let's count the number of electrons with mℓ=1 in p and f orbitals: - In p orbitals: * 1p (mℓ=1): 0 * 2p (mℓ=1): 2 * 3p (mℓ=1): 2 * 4p (mℓ=1): 2 * 5p (mℓ=1): 2 - In f orbitals: * 4f (mℓ=1): 2 * 5f (mℓ=1): 2 The total number of electrons with mℓ=1 is 2+2+2+2+2+2=12 electrons. #d. Electrons with ms=-1/2#

Determine the number of electrons with ms=-1/2

In an atom with Z protons (atomic number), there will be Z electrons. Since it is neutral and in its ground state, element 115 has 115 electrons. Half of these electrons will have ms=-1/2, and the other half will have ms=1/2. Therefore, the total number of electrons with ms=-1/2 is 115/2 = 57.5, but since electrons cannot be in halves, we round up to the nearest whole number, which is 58 electrons.